Number System Fundamentals • Topic 2 of 7

Number Properties

A few properties of whole numbers cut down a lot of work. The sum of two odd numbers or two even numbers is even, while odd + even is odd. The product of consecutive integers is always even, and the product of any n consecutive integers is divisible by n!. The sum of the first n natural numbers is n(n+1)/2. A perfect square can only end in 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8. A two-digit number minus its reverse is always a multiple of 9. Spotting these patterns replaces long computation with one-line reasoning.

✅ Solved examples

1. If a and b are both odd, is a + b odd or even?
The sum of two odd numbers is always even. For example 3 + 5 = 8. So a + b is even.
2. The product of four consecutive integers is always divisible by what largest number?
Among any four consecutive integers, the product is divisible by 4! = 4 × 3 × 2 × 1 = 24. So the largest guaranteed divisor is 24.
3. Find the sum of the first 50 natural numbers.
Use n(n+1)/2 with n = 50: 50 × 51 / 2 = 2550 / 2 = 1275.
4. Can a perfect square end in the digit 7?
Squares can only end in 0, 1, 4, 5, 6 or 9. The digit 7 never appears as the last digit of a perfect square, so no.

✏️ Practice — try these, take hints as needed

1. If n is an even number, is n(n + 1) even or odd?
One of n and n+1 is even.
Here n itself is even.
An even number times anything is even.
Even.
2. Find the sum of the first 30 natural numbers.
Use the formula n(n+1)/2.
Substitute n = 30.
Compute 30 × 31 / 2.
465.
3. The product of three consecutive integers is always divisible by which number?
Think about how many consecutive integers you have.
For n consecutive integers the product is divisible by n!.
Here n = 3, so use 3!.
6.
4. A two-digit number minus the number formed by reversing its digits is always divisible by?
Write the number as 10t + u and its reverse as 10u + t.
Subtract: (10t + u) − (10u + t) = 9t − 9u.
Factor out 9.
9.
5. Which of these can be the units digit of a perfect square: 2, 3, 7, or 9?
Square the digits 0–9 and look at the last digits.
Possible last digits are 0, 1, 4, 5, 6, 9 only.
Pick the one option that appears in that list.
9 (e.g. 3² = 9).

📝 Topic test — 8 questions

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